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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"4628972c41f68ece6d2fd7606c0a960d9ee646260a8fc40bc8979c2f43d0bcc1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-57b4b14604f4c0d7","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:     1;\n  1:     3,      1;\n  2:    10,      7,      1;\n  3:    35,     39,     12,      1;\n  4:   126,    196,     98,     18,      1;\n  5:   462,    930,    660,    200,     25,     1;\n  6:  1716,   4257,   3960,   1725,    360,    33,     1;\n  7:  6435,  19019,  22022,  12705,   3850,   595,    42,    1;\n  8: 24310,  83512, 116116,  84084,  34034,  7700,   924,   52,  1;\n  9: 92378, 361998, 588744, 515424, 263718, 80262, 14196, 1368, 63, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^3+3*n^2+2*n+(2*n^2+3*n-2)*x+(n+2)*x^2)*P(n,x) = (4*n^3+10*n^2+4*n+(3*n^3+13*n^2+n-2)*x+(6*n^2+4*n+2)*x^2+3*n*x^3)*P(n-1,x) - ((2*n^2+5*n+2)*x+(2*n^2+9*n+4)*x^2+(4*n+4)*x^3+2*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^3 + 10*n^2 + 4*n + (3*n^3 + 13*n^2 + n - 2)*x + (6*n^2 + 4*n + 2)*x^2 + 3*n*x^3)*P[n - 1, x] - ((2*n^2 + 5*n + 2)*x + (2*n^2 + 9*n + 4)*x^2 + (4*n + 4)*x^3 + 2*x^4)*D[P[n - 1, x], x])/(n^3 + 3*n^2 + 2*n + (2*n^2 + 3*n - 2)*x + (n + 2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^3 + 10*n^2 + 4*n + (3*n^3 + 13*n^2 + n - 2)*x + (6*n^2 + 4*n + 2)*x^2 + 3*n*x^3)*v[i - 1] - ((2*n^2 + 5*n + 2)*x + (2*n^2 + 9*n + 4)*x^2 + (4*n + 4)*x^3 + 2*x^4)*deriv(v[i - 1], x))/(n^3 + 3*n^2 + 2*n + (2*n^2 + 3*n - 2)*x + (n + 2)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-57b4b14604f4c0d7","steps":["D+1"],"edges":["e-49264023c59aa288"],"text":"A126216 \u2192 D+1"},"headline":"A126216 \u2192 D+1","chain":null}
